Planar velocity dynamos in a sphere

2006 ◽  
Vol 462 (2072) ◽  
pp. 2439-2456 ◽  
Author(s):  
A.A Bachtiar ◽  
D.J Ivers ◽  
R.W James
Keyword(s):  
Magnetic Field ◽  
Outer Core ◽  
Magnetic Reynolds Number ◽  
Main Magnetic Field ◽  
Dynamo Action ◽  
Field Growth ◽  
Planar Velocity ◽  
Induction Equation ◽  
Planar Flows ◽  
Magnetic Induction Equation

The Earth's main magnetic field is generally believed to be due to a self-exciting dynamo process in the Earth's fluid outer core. A variety of antidynamo theorems exist that set conditions under which a magnetic field cannot be indefinitely maintained by dynamo action against ohmic decay. One such theorem, the Planar Velocity Antidynamo Theorem , precludes field maintenance when the flow is everywhere parallel to some plane, e.g. the equatorial plane. This paper shows that the proof of the Planar Velocity Theorem fails when the flow is confined to a sphere, due to diffusive coupling at the boundary. Then, the theorem reverts to a conjecture. There is a need to either prove the conjecture, or find a functioning planar velocity dynamo. To the latter end, this paper formulates the toroidal–poloidal spectral form of the magnetic induction equation for planar flows, as a basis for a numerical investigation. We have thereby determined magnetic field growth rates associated with various planar flows in spheres. For most flows, the induced magnetic field decays with time, supporting a planar velocity antidynamo conjecture for a spherical conducting fluid. However, one flow is exceptional, indicating that magnetic field growth can occur. We also re-examine some classical kinematic dynamo models, converting the flows where possible to planar flows. For the flow of Pekeris et al . (Pekeris, C. L., Accad, Y. & Shkoller, B. 1973 Kinematic dynamos and the Earth's magnetic field. Phil. Trans. R. Soc. A 275 , 425–461), this conversion dramatically reduces the critical magnetic Reynolds number.

Kinematic dynamo action in a network of screw motions; application to the core of a fast breeder reactor

1999 ◽  
Vol 382 ◽  
pp. 137-154 ◽  
Author(s):  
F. PLUNIAN ◽  
P. MARTY ◽  
A. ALEMANY
Keyword(s):  
Magnetic Field ◽  
Experimental Studies ◽  
Magnetic Reynolds Number ◽  
Liquid Sodium ◽  
Dynamo Action ◽  
Periodic Cell ◽  
The Core ◽  
Induction Equation ◽  
Breeder Reactors ◽  
Dynamo Effect

Most of the studies concerning the dynamo effect are motivated by astrophysical and geophysical applications. The dynamo effect is also the subject of some experimental studies in fast breeder reactors (FBR) for they contain liquid sodium in motion with magnetic Reynolds numbers larger than unity. In this paper, we are concerned with the flow of sodium inside the core of an FBR, characterized by a strong helicity. The sodium in the core flows through a network of vertical cylinders. In each cylinder assembly, the flow can be approximated by a smooth upwards helical motion with no-slip conditions at the boundary. As the core contains a large number of assemblies, the global flow is considered to be two-dimensionally periodic. We investigate the self-excitation of a two-dimensionally periodic magnetic field using an instability analysis of the induction equation which leads to an eigenvalue problem. Advantage is taken of the flow symmetries to reduce the size of the problem. The growth rate of the magnetic field is found as a function of the flow pitch, the magnetic Reynolds number (Rm) and the vertical magnetic wavenumber (k). An α-effect is shown to operate for moderate values of Rm, supporting a mean magnetic field. The large-Rm limit is investigated numerically. It is found that α=O(Rm−2/3), which can be explained through appropriate dynamo mechanisms. Either a smooth Ponomarenko or a Roberts type of dynamo is operating in each periodic cell, depending on k. The standard power regime of an industrial FPBR is found to be subcritical.

Numerical solutions of the kinematic dynamo problem

1973 ◽  
Vol 274 (1241) ◽  
pp. 493-521 ◽  
Keyword(s):  
Magnetic Field ◽  
Numerical Solutions ◽  
Expansion Method ◽  
Magnetic Reynolds Number ◽  
Axially Symmetric ◽  
Dynamo Action ◽  
Rotating Magnetic Fields ◽  
Induction Equation ◽  
Steady Solutions ◽  
The Magnetic Field

The expansion method of Bullard & Gellman is used to find numerical solutions of the induction equation in a sphere of conducting fluid. Modifications are made to the numerical methods, and one change due to G. O. Roberts greatly increases the efficiency of the scheme. Calculations performed recently by Lilley are re-examined. His solutions, which appeared to be convergent, are shown to diverge when a higher level of truncation is used. Other similar dynamo models are investigated and it is found that these also do not provide satisfactory steady solutions for the magnetic field. Axially symmetric motions which depend on spherical harmonics of degree n are examined. Growing solutions, varying with longitude, 0, as e1^, are found for the magnetic field, and numerical convergence of the solutions is established. The field is predominantly an equatorial dipole with a toroidal field symmetric about the same axis. When n is large the problem lends itself to a two-scale analysis. Comparisons are made between the approximate results of the two-scale method and the numerical results. There is agreement when n is large. When n is small the efficiency of the dynamo is lowered. It is shown that the dominant effect of a large microscale magnetic Reynolds number is the expulsion of magnetic flux by eddies to give a rope-like structure for part of the field. Physical interpretations are given which explain the dynamo action of these motions, and of related flows which support rotating magnetic fields.

Turbulent dynamo action at low magnetic Reynolds number

1970 ◽  
Vol 41 (2) ◽  
pp. 435-452 ◽  
Author(s):  
H. K. Moffatt
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Magnetic Energy ◽  
Magnetic Reynolds Number ◽  
Dynamo Action ◽  
Ohmic Dissipation ◽  
Magnetic Energy Density ◽  
Magnetic Diffusivity

The effect of turbulence on a magnetic field whose length-scale L is initially large compared with the scale l of the turbulence is considered. There are no external sources for the field, and in the absence of turbulence it decays by ohmic dissipation. It is assumed that the magnetic Reynolds number Rm = u0l/λ (where u0 is the root-mean-square velocity and λ the magnetic diffusivity) is small. It is shown that to lowest order in the small quantities l/L and Rm, isotropic turbulence has no effect on the large-scale field; but that turbulence that lacks reflexional symmetry is capable of amplifying Fourier components of the field on length scales of order Rm−2l and greater. In the case of turbulence whose statistical properties are invariant under rotation of the axes of reference, but not under reflexions in a point, it is shown that the magnetic energy density of a magnetic field which is initially a homogeneous random function of position with a particularly simple spectrum ultimately increases as t−½exp (α2t/2λ3) where α(= O(u02l)) is a certain linear functional of the spectrum tensor of the turbulence. An analogous result is obtained for an initially localized field.

scholarly journals Torsional waves driven by convection and jets in Earth’s liquid core

2018 ◽  
Vol 216 (1) ◽  
pp. 123-129 ◽  
Author(s):  
R J Teed ◽  
C A Jones ◽  
S M Tobias
Keyword(s):  
Magnetic Field ◽  
Density Gradient ◽  
Inner Core ◽  
Liquid Core ◽  
Outer Core ◽  
Dynamo Action ◽  
Radial Magnetic Field ◽  
Torsional Waves ◽  
Plausible Mechanism ◽  
Rich Liquid

SUMMARY Turbulence and waves in Earth’s iron-rich liquid outer core are believed to be responsible for the generation of the geomagnetic field via dynamo action. When waves break upon the mantle they cause a shift in the rotation rate of Earth’s solid exterior and contribute to variations in the length-of-day on a ∼6-yr timescale. Though the outer core cannot be probed by direct observation, such torsional waves are believed to propagate along Earth’s radial magnetic field, but as yet no self-consistent mechanism for their generation has been determined. Here we provide evidence of a realistic physical excitation mechanism for torsional waves observed in numerical simulations. We find that inefficient convection above and below the solid inner core traps buoyant fluid forming a density gradient between pole and equator, similar to that observed in Earth’s atmosphere. Consequently, a shearing jet stream—a ‘thermal wind’—is formed near the inner core; evidence of such a jet has recently been found. Owing to the sharp density gradient and influence of magnetic field, convection at this location is able to operate with the turnover frequency required to generate waves. Amplified by the jet it then triggers a train of oscillations. Our results demonstrate a plausible mechanism for generating torsional waves under Earth-like conditions and thus further cement their importance for Earth’s core dynamics.

scholarly journals Large-scale dynamo action of magnetized Taylor–Couette flows

2020 ◽  
Vol 493 (1) ◽  
pp. 1249-1260
Author(s):  
G Rüdiger ◽  
M Schultz
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Mean Field ◽  
Single Mode ◽  
Eddy Diffusivity ◽  
Magnetic Reynolds Number ◽  
Dynamo Model ◽  
Turbulence Energy ◽  
Dynamo Action ◽  
Taylor Couette

ABSTRACT A conducting Taylor–Couette flow with quasi-Keplerian rotation law containing a toroidal magnetic field serves as a mean-field dynamo model of the Tayler–Spruit type. The flows are unstable against non-axisymmetric perturbations which form electromotive forces defining α effect and eddy diffusivity. If both degenerated modes with m = ±1 are excited with the same power then the global α effect vanishes and a dynamo cannot work. It is shown, however, that the Tayler instability produces finite α effects if only an isolated mode is considered but this intrinsic helicity of the single-mode is too low for an α2 dynamo. Moreover, an αΩ dynamo model with quasi-Keplerian rotation requires a minimum magnetic Reynolds number of rotation of Rm ≃ 2000 to work. Whether it really works depends on assumptions about the turbulence energy. For a steeper-than-quadratic dependence of the turbulence intensity on the magnetic field, however, dynamos are only excited if the resulting magnetic eddy diffusivity approximates its microscopic value, ηT ≃ η. By basically lower or larger eddy diffusivities the dynamo instability is suppressed.

Experimental investigation of dynamo effect in the secondary pumps of the fast breeder reactor Superphenix

2000 ◽  
Vol 403 ◽  
pp. 263-276 ◽  
Author(s):  
A. ALEMANY ◽  
Ph. MARTY ◽  
F. PLUNIAN ◽  
J. SOTO
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Magnetic Energy ◽  
Experimental Studies ◽  
Phenomenological Approach ◽  
Magnetic Reynolds Number ◽  
Experimental Investigations ◽  
Dynamo Action ◽  
Experimental Conditions ◽  
Dynamo Effect

The fast breeder reactors (FBR) BN600 (Russia) and Phenix (France) have been the subject of several experimental studies aimed at the observation of dynamo action. Though no dynamo effect has been identified, the possibility was raised for the FBR Superphenix (France) which has an electric power twice that of BN600 and five times larger than Phenix. We present the results of a series of experimental investigations on the secondary pumps of Superphenix. The helical sodium flow inside one pump corresponds to a maximum magnetic Reynolds number (Rm) of 25 in the experimental conditions (low temperature). The magnetic field was recorded in the vicinity of the pumps and no dynamo action has been identified. An estimate of the critical flow rate necessary to reach dynamo action has been found, showing that the pumps are far from producing dynamo action. The magnetic energy spectrum was also recorded and analysed. It is of the form k−11/3, suggesting the existence of a large-scale magnetic field. Following Moffatt (1978), this spectrum slope is also justified by a phenomenological approach.

scholarly journals General relativistic magnetic perturbations and dynamo effects in extragalactic radiosources

2010 ◽  
Vol 6 (S274) ◽  
pp. 393-397
Author(s):  
L. C. Garcia de Andrade
Keyword(s):  
Magnetic Field ◽  
Magnetic Energy ◽  
Magnetic Fluctuations ◽  
Dynamo Action ◽  
Magnetic Contrast ◽  
Mhd Dynamo ◽  
Curvature Energy ◽  
Induction Equation ◽  
The Universe ◽  
The Magnetic Field

AbstractBy making use of the MHD self-induction equation in general relativity (GR), recently derived by Clarkson and Marklund (2005), it is shown that when Friedmann universe possesses a spatial section whose Riemannian curvature is negative, the magnetic energy bounds computed by Nuñez (2002) also bounds the growth rate of the magnetic field given by the strain matrix of dynamo flow. Since in GR-MHD dynamo equation, the Ricci tensor couples with the universe magnetic field, only through diffusion, and most ages are highly conductive the interest is more theoretical here, and only very specific plasma astrophysical problems can be address such as in laboratory plasmas. Magnetic fields and the negative curvature of some isotropic cosmologies, contribute to enhence the amplification of the magnetic field. Ricci curvature energy is shown to add to strain matrix of the flow, to enhance dynamo action in the universe. Magnetic fluctuations of the Clarkson-Marklund equations for a constant magnetic field seed in highly conductive flat universes, leads to a magnetic contrast of ≈ 2, which is well within observational limits from extragalactic radiosources of ≈ 1.7. In the magnetic helicity fluctuations the magnetic contrast shows that the dynamo effects can be driven by these fluctuations.

A mapping method for solving dynamo equations

1995 ◽  
Vol 448 (1932) ◽  
pp. 1-28 ◽  
Keyword(s):  
Magnetic Field ◽  
Decay Rate ◽  
Mapping Method ◽  
Three Dimensions ◽  
New Method ◽  
Dynamo Model ◽  
Dynamo Action ◽  
Electrically Conducting ◽  
Induction Equation ◽  
Conducting Fluids

A new method is presented for integration of the equations of magnetohydrody­namics in rapidly rotating, electrically conducting fluids, and in particular for studying dynamo action in such systems. Tests of the method are reported. The decay rate of magnetic field in a stationary sphere are recovered, as are the results of a number of α 2 - and αω -dynamos. These are solutions of the electrodynamic (induction) equation. An intermediate dynamo model, in which the dynamics are also partly allowed for, is also studied. This is due to Braginsky ( Geomag. Aero­naut . 18, 225 (1978)) and is of ‘model- Z type’. All models considered here are axisymmetric, but the possibility of generalization to three-dimensions is allowed for.

Dimensionality of Anisotropic Homogeneous Turbulence With Distortions: Compared Effects Between the Coriolis Force and the Lorentz Force

2014 ◽  
Author(s):  
Fabien Godeferd ◽  
Claude Cambon ◽  
Alexandre Delache
Keyword(s):  
Magnetic Field ◽  
Lorentz Force ◽  
Coriolis Force ◽  
Magnetic Reynolds Number ◽  
Homogeneous Turbulence ◽  
Spectral Space ◽  
Structure And Dynamics ◽  
Induction Equation ◽  
The Magnetic Field

We consider initially isotropic homogeneous turbulence which is submitted to an external force, in statistically axisymmetric configurations. First, we study hydrodynamical turbulence in a rotating frame, in which case the Coriolis force modifies the structure and dynamics of the flow, thus creating elongated structures along the axis of rotation, corresponding to an accumulation of energy in the neighbourhood of the equatorial spectral plane. Secondly, a very similar configuration is that of magnetohydrodynamics (MHD) of a conducting fluid within an externally applied space uniform magnetic field, in which case the Lorentz force also concentrates energy to the same spectral equatorial manifold, but creates axially extending current sheets, along the magnetic field. We more specifically consider the quasi-static limit at small magnetic Reynolds number, in which the induction equation is analytically solved. We study the anisotropy of each turbulent flow using progressively refined statistics applied to results of direct numerical simulations, and we show that an accurate characterization of the flow structure requires advanced two-point statistics, which are available easily only in spectral space.

scholarly journals Nonlinear Dynamo of Magnetic Fluctuations and Flux Tubes Formation in the Ionosphere of Venus

1993 ◽  
Vol 157 ◽  
pp. 481-486
Author(s):  
N. Kleeorin ◽  
I. Rogachevskii ◽  
A. Eviatar
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Magnetic Reynolds Number ◽  
Small Scale ◽  
Magnetic Fluctuations ◽  
Flux Tubes ◽  
Magnetic Flux Ropes ◽  
Flux Ropes ◽  
Induction Equation ◽  
The Magnetic Field

Magnetic field observations in the dayside ionosphere of Venus revealed the magnetic flux ropes (Russell and Elphic 1979). General properties of these small-scale magnetic field structures can be explained by the theory of magnetic fluctuations excited by random hydrodynamic flows of ionospheric plasma.A nonlinear theory of the flux tubes formation based on the Zeldovich's mechanism of amplification of the magnetic fluctuations is proposed. A nonlinear equation describing the evolution of the correlation function of the magnetic field can be derived from the induction equation, the nonlinearity being connected with the Hall effect. The large magnetic Reynolds number limit allows an asymptotic study by a modified WKB method.On the basis of this theory it is possible to explain why the flux tubes are not observed if there is a strong regular large-scale magnetic field when the ionopause is low. The theory predicts the cross section of the flux ropes in the ionosphere of Venus and the maximum value of the magnetic field inside the flux tube.

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